Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61569 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | #3505 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
881 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,26)(20,24)(21,29)(22,25)(23,28)(27,32)(30,31);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(19,27)(23,31)(26,32)(28,30);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19,22,27)(18,23,25,31)(20,28,21,30)(24,32,29,26);(3,11)(6,9)(10,16)(13,15)(17,20,22,21)(18,24,25,29)(19,28,27,30)(23,32,31,26);(2,8)(6,9)(7,14)(13,15)Polycyclic presentation:
2^8|c^2=fh,d^2=h,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,c^b=ch,d^b=df,g^c=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | #245677, #245680, #245701, … |
| maximal normal subgroup of | #245677, #245680, #245701, … |
| maximal quotient of | #6386147, #6387806, #6387814, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54700);